The finite option to invest is investigated using Monte Carlo simulation for American options.
Due to long computational time only 5000 simulations are used. The resulting values from 5000 simulations do not show large variations, and is considered to provide the accuracy needed for the investment decision as shown in Table 4.
Model 3 is governed by two unobservable stochastic processes. The short term process shows strong mean reversion, a property which makes the long term variable dominant when it comes to value the investment and the option. Short-term fluctuations in the “spot” price do only to a certain degree have a positive effect on the value. Figure 21 displays the spark spread “spot” price and the two state variables. The long term variable is less volatile than the short term.
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"Spot" and state variables
-200,00 -100,00 0,00 100,00 200,00 300,00 400,00 500,00 600,00 700,00
21.01.2000 20.01.2001 20.01.2002 20.01.2003
NOK/MWh
Spot Short term state variable Long term state variable
Figure 21 “Spot” and state variables – Model 3
Under the base case scenario model 3 gives an option value of 3537.4 MNOK and a NPV of 2862.3 MNOK. This calls for delaying the investment.
Performing sensitivity analysis on the parameters related to the short-term stochastic process shows that changes in these parameters have very little effect on the valuation. κ, is assumed to be greater than the mean reverting coefficient of model 2. The short-term volatility can take on almost any value without making any changes the values of option and investment.
Sensitivity analysis performed on the risk free drift parameter µ* in the long-term stochastic process states that this variable is very important to the option and investment value. Figure 22 shows how the option- and investment value vary with the yearly risk free drift. Changes in the drift will not alter the investment decision since the exercise value never exceeds the option value. The lines in Figure 22 and Figure 23 are not straight due to the inaccuracy in Monte Carlo simulation technique.
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1000 2000 3000 4000 5000 6000 7000 8000
0 2 4 µ∗ 6 8 10
MNOK
Option value NPV of investment
Estimated value
Figure 22: Varying risk free drift – Model 3
The value ofS, the long-term average of the historic market data, has a major impact on the investment decision. Increasing this parameter in the range 0 NOK/MWh to 60 NOK/MWh, Figure 23 shows how the option value closes in on the investment value. If Sis increased by more then about 56 NOK/MWh, an investment should commence. Since the mean reversion coefficient is very strong the difference in increasing α*instead of Sis negligible. Thus increasingα*, which reflects the long-term expected price level, with about 56 NOK/MWh shows the same result. The uncertainty lays in α*where the estimation had a RMSE of 13 NOK/MWh. In comparison α* and Sare 12.3 NOK/MWh and 83.4 NOK/MWh respectively as given in Table 8.
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1000 2000 3000 4000 5000 6000 7000 8000
0 10 20 30 40 50 60
Increase in a* or SS in NOK/MWh
MNOK
Option value NPV of investment Investment threshold
Figure 23: Option value and NPV as a function of an increase in α* or S – Model 3
Varying the risk free interest rate in the range from 2%-10% per year have a large impact on the values of both investment and the option, but the investment decision is not altered.
Similar results appear when varying the price of CO2 quotas. The price per quota does not seem to revise the investment decision except when the expected price of CO2 is zero. In general the conclusion that investment should be delayed is robust.
To say something about how the option value varies with the “spot” price is not straight forward since there is no one-to-one relation between the state variables and the spark spread
“spot”. One procedure could be to find spot prices in the historical data and then use the equivalent state variables found in the parameter estimation for the given date to do the calculation. This however, is not a consistent procedure. Valuing the option when the “spot”
price increased in December 2002 gives higher values than valuing the option at the same price when the “spot” price decreased during the winter of 2003 because the long-term variable had a higher value in the December 2002 than during the following winter for equivalent “spot” prices.
The investment decision is hardly affected by the short term fluctuations. An alternative model could be to remove the terms in (8.46) that originate from the short term process exceptα*, since it contains future market information, and let the stochastic process be given by (8.43). This is will be a model very similar to model 1 but with St =S+εt +α* in the
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equation (8.4) of the forward price. Analysing the investment decision using the outlined model also calls for delaying the investment decision. The investment trigger of an infinite option, which is very similar to an option with 10 years to maturity, is 144 NOK/MWh.
Subtracting 7ε0 =−7. NOK/MWh and α* =12.3NOK/MWh from this threshold requires S=140 NOK/MWh which is the same result as in the sensitivity analysis of S. This will be referred to as model 3b.
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11 Discussion of the analyses
The main results from chapter 10 are summarized in Table 9. The models do not consent on a clear investment decision. In this chapter the underlying causes for these differences will be explained and analyzed, in order to make a choice of the best model. The parameters that are most likely to alter the decisions are also summarized.
Model
Option value MNOK
Investment threshold NOK/MWh
Equivalent spot price*
NOK/MWh
NPV of investment
MNOK
Investment decision
1 3499 144 86 2463 "Wait"
2 0 N/A** 195 -1280 "Never invest"
3a 3540 N/A*** 195 2860 "Wait"
3b 3580 144 88 2600 "Wait"
Table 9: *For model 2 and 3a this is the spark spread “spot” reflected by the constructed “spot” price as described in chapter 5.2. In model 1 and 3b the “spot” price is some price determined by the long term information in the market as described in 9.2 for model 1 and 10.3 for model 3b. ** Since the option never will take on any value no exercise boundary can be found. *** Model 3a has two parameters and the exercise boundary will be a line in the [X, ε] space.